Pendulum mechanical energy at different amplitudes

[mikefitz]

A pendulum of length 130 cm swings with an amplitude of 2.1 cm. Its mechanical energy is 7 mJ. What is the mechanical energy of the same pendulum when it swings with an amplitude of 3.9 cm?

Well I know that energy is conserved in such a system; what I don't know is how to calculate KE and PE of this pendulum (also, I know that mass is independent of frequency). I've read that total ME is proportional to the square of the amplitude, though I'm not sure how to work that into my solution.

My book does have a chapter on simple pendulums, with equations for calculating the period, frequency, and angular frequency - what it does not have are equations for calculating a pendulums energy. Any help? thanks

 

[OlderDan]

A pendulum of length 130 cm swings with an amplitude of 2.1 cm. Its mechanical energy is 7 mJ. What is the mechanical energy of the same pendulum when it swings with an amplitude of 3.9 cm?

Well I know that energy is conserved in such a system; what I don't know is how to calculate KE and PE of this pendulum (also, I know that mass is independent of frequency). I've read that total ME is proportional to the square of the amplitude, though I'm not sure how to work that into my solution.

My book does have a chapter on simple pendulums, with equations for calculating the period, frequency, and angular frequency - what it does not have are equations for calculating a pendulums energy. Any help? thanks

The stored energy in a swinging pendulum is the gravitational potential energy (GPE) of the bob. When the bob is not moving, all the energy is GPE. For any harmonic oscillator it is true that the total energy is proportional to the square of the amplitude. A pendulum is very nearly a hamonic oscillator, as long as the angles are small and the small angle approximation is valid. You shouild expect a calculation of the energy based on GPE to be consistent with the general behavior of harmonic oscillators.

 

[mikefitz]

OlderDan, I really don't know how to translate that thought into an equation to solve this problem; can you explain it in a way that would me it easier for me to visualize what is going on? Thanks

 

[OlderDan]

OlderDan, I really don't know how to translate that thought into an equation to solve this problem; can you explain it in a way that would me it easier for me to visualize what is going on? Thanks

Let the zero of GPE be at the lowest point of the pendulum's swing. As it swings to either side, the height of the bob increases relative to this lowest position. When the bob is displaced 2.1cm and released it will rise to its maximum height (for that initial displacement) above the lowest point and its total energy will be the GPE associated with that rise. Knowing the energy for this displacement will enable you to determine the mass of the bob. When it is displaced 3.9cm it will have a higher maximum height and a higher total energy. You only need to compare the heights associated with these two displacements and the associated GPEs to answer the question.